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The unique factorization property is a direct consequence of [[Euclid's lemma]]: If an [[irreducible element]] divides a product, then it divides one of the factors. For univariate polynomials over a field, this results from [[Bézout's identity]], which itself results from [[Euclidean algorithm]].
So, let {{math|''R''}} be a unique factorization ___domain, which is not a field, and {{math|''R''[''X'']}} the univariate [[polynomial ring]] over {{math|''R''}}. An irreducible element {{math|''r''}} in {{math|''R''[''X'']}} is either an irreducible element in {{math|''R''}} or an irreducible primitive polynomial.
If {{math|''r''}} is in {{math|''R''}} and divides a product <math>P_1P_2</math> of two polynomials, then it divides the content <math>c(P_1P_2) = c(P_1)c(P_2).</math> Thus, by Euclid's lemma in {{math|''R''}}, it divides one of the contents, and therefore one of the polynomials.
If {{math|''r''}} is not {{math|''R''}}, it is a primitive polynomial (because it is irreducible). Then Euclid's lemma in {{math|''R''[''X'']}} results immediately from Euclid's lemma in {{math|''K''[''X'']}}, where {{math|''K''}} is the field of fractions of {{math|''R''}}.
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